Stability and Bifurcations of Equilibria in Networks with Piecewise Linear Interactions

In this paper, we study equilibria of differential equation models for networks. When interactions between nodes are taken to be piecewise constant, an efficient combinatorial analysis can used characterize the equilibria. constant functions replaced with linear functions, preserved as long sufficiently steep. Therefore leveraged understand a broader class interactions. To better how broad is, explicitly steep must correspondence hold. do so, analyze steady state and Hopf bifurcations which cause change in number or stability slopes decreased. Additionally, show choose subset parameters so that holds smallest possible when remaining fixed.